What is the Derivative of Sin(x)?
we are going to share formula f derivative of sin (x) and proof .
\( \frac{d}{dx} sin (x) = \lim_{d \to 0} \frac{(sin(x+d)- sin(x))}{d} \)
You may remember the following angle sum formula from high school: \( sin(a+b) = sin(a)cos(b) + sin(b)cos(a) \)
This lets us untangle the x from the d as follows:
\( \frac{d}{dx} sin (x) = \lim_{d \to 0} \frac{(sin(x)cos(d) + sin(d)cos(x)- sin(x))}{d} \)
We can simplify this expression using some basic algebraic facts:
=> \( \frac{d}{dx} sin (x) = \lim_{d \to 0} [\frac{sin(x)cos(d) – sin(x)}{d} + \frac{sin(d)cos(x)}{d} ] \)
=> \( \frac{d}{dx} sin (x) = \lim_{d \to 0} [ sin(x) (\frac {cos(d) – 1}{d} ) + cos(x) \frac{sin(d)}{d} ] \)
=> \( \frac{d}{dx} sin (x) = \lim_{d \to 0} sin(x) (\frac {cos(d) – 1}{d} ) + \lim_{d \to 0} cos(x) (\frac{sin(d)}{d}) \)
=> \( \frac{d}{dx} sin (x) = sin(x) \lim_{d \to 0} (\frac {cos(d) – 1}{d} ) + cos(x) \lim_{d \to 0} (\frac{sin(d)}{d}) \)
We Know that
\( \lim_{d \to 0} (\frac {cos(d) – 1}{d} ) =0 \)
\( \lim_{d \to 0} (\frac{sin(d)}{d} ) = 1 \)
put the Value of \( \lim_{d \to 0} (\frac {cos(d) – 1}{d} ) =0 \) & \( \lim_{d \to 0} (\frac{sin(d)}{d} ) = 1 \)
=> \( \frac{d}{dx} sin (x) = sin(x) \times 0 + cos(x) \times 1 \)
=> \( \frac{d}{dx} sin (x) = cos(x) \)
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